12.1 Mathematical Model

In order to study the behaviour of a mechanical system consisting of a cable with its shaft, a gear lever and its support (see in Fig. 12.1) we introduce a simplified seven degrees of freedom (DOF) system. An equivalent mass m with displacement X ₄ and clearance ? exhibits impact nonlinearity due to the motor (see Fig. 12.2). Though a real physical external excitation would be more complex, we only consider a simple form of excitation u(t)= ?sin ( ?t) with amplitude ? and frequency ?.

Fig. 12.1: Simplified model of the gear-box.

Fig. 12.2: Simplified model of the cable with clearance contact on the motor side.

The cable is represented by a mass M ₁ with displacement X ₅ and equivalent stiffness k ₁ (see Fig. 12.2): this is a one DOF reduction of the continuous system. Impacts between m and M ₁ are governed by a classical restitution law with coefficient e. The displacements of masses m and M ₁ are governed by the equations:

(12.1)

(12.2)

The functions X ₄ and X ₅ remain smooth if X ₄ ? X ₅ ?] ? ?, ?[. Elastic impacts occur if X ₄ ? X ₅= ?. Using some algebra (related to momentum conservation [Brogliato (1996)]) velocities and after an impact are given by the relations:

(12.3)

where and